Advertisement

Doklady Physics

, Volume 63, Issue 7, pp 297–301 | Cite as

Stability in the Regular Precession of an Asymmetrical Gyroscope in the Critical Case of Fourth-Order Resonance

  • A. P. Markeev
MECHANICS
  • 3 Downloads

Abstract

The motion of a solid around a stationary point in a uniform gravity field is considered. The mass geometry of this body is such that it can perform regular precession around an axis inclined to a vertical (Grioli precession). The problem about the orbital stability of this precession is solved in the critical case of fourth-order resonance, when the terms to a power higher than the fourth with respect to perturbations (including the sixth-power terms) must be taken into account in the expansion of the Hamiltonian function.

Notes

REFERENCES

  1. 1.
    G. Grioli, Ann. Math. Pura Appl. 26 (3), 271 (1947).CrossRefGoogle Scholar
  2. 2.
    M. P. Gulyaev, Vestn. Mosk. Univ., Ser. Fiz.-Mat. Estestv. Nauk 3, 15 (1955).Google Scholar
  3. 3.
    G. Grioli, Ann. Sc. Norm. Super. Pisa 1 (3), 43 (1949).Google Scholar
  4. 4.
    A. Z. Bryum, Mekh. Tverd. Tela 19, 68 (1987).Google Scholar
  5. 5.
    G. V. Mozalevskaya, A. P. Kharlamov, and E. I. Kharlamova, Mekh. Tverd. Tela 24, 15 (1992).Google Scholar
  6. 6.
    V. N. Tkhai, J. Appl. Math. Mech. 64 (5), 811 (2000).CrossRefGoogle Scholar
  7. 7.
    A. P. Markeyev, J. Appl. Math. Mech. 66 (6), 889 (2002).MathSciNetCrossRefGoogle Scholar
  8. 8.
    A. P. Markeev, Dokl. Akad. Nauk 387 (3), 338 (2002).MathSciNetGoogle Scholar
  9. 9.
    A. P. Markeyev, J. Appl. Math. Mech. 67 (4), 497 (2003).MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. P. Markeev, Regul. Chaotic Dyn. 22 (7), 773 (2017).ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    E. T. Whittaker, Analytical Dynamics (Cambridge University Press, Cambridge, 1927).MATHGoogle Scholar
  12. 12.
    A. P. Markeyev, J. Appl. Math. Mech. 61 (3), 355 (1997).MathSciNetCrossRefGoogle Scholar
  13. 13.
    A. P. Markeyev, J. Appl. Math. Mech. 78 (5), 435 (2014).MathSciNetCrossRefGoogle Scholar
  14. 14.
    A. P. Markeev, Nelineinaya Din. 11 (3), 503 (2015).CrossRefGoogle Scholar
  15. 15.
    Yu. A. Arkhangel’skii, Analytical Dynamics of Solids (Nauka, Moscow, 1977) [in Russian].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Ishlinsky Institute for Problems in Mechanics RASMoscowRussia
  2. 2.Moscow Aviation Institute (National Research University)MoscowRussia
  3. 3.Moscow Institute of Physics and Technology (State University)DolgoprudnyiRussia

Personalised recommendations